Summer School 2001:
Homological conjectures for finite dimensional algebras

The lectures represent joint work with S. O. Smalø. The first of the two lectures will deal with the relevance of phantoms in testing given categories of representations for contravariant finiteness and in replacing part of the information held by minimal approximations in case of failure of contravariant finiteness.

The second talk will make the subject more concrete by showing that a finite list of phantoms encodes the homology of string algebras; in particular, these (not necessarily finite dimensional) representations provide an algorithmic tool for testing whether the category of finitely generated modules of finite projective dimension is contravariantly finite in the full category of finitely generated representations.